C₅×#71095 #14316442

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#71095

Splittings

Fitting	C₃×C₅×#472 ⋊ C₂
Hall	C₃×C₅ ⋊ #2193
Sylow	C₃ ⋊ C₅×#2193

Subgroups

	subgroup	G/H
center	C₂×C₅	#5036
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#472	C₂
socle	C₂×C₃×C₅	#447
hypercenter	C₅×#472	D₃
hypocenter	C₃	C₅×#2193
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#131 ◁ C₅×#472

420 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(6,7,8)(9,11,15,22)(10,17,19,34)(12,21,26,37)(13,27,14,20)(16,29,30,40)(18,33,28,32)(23,36,25,39)(24,38,35,31);(7,8)(9,10)(11,17)(12,24)(13,18)(14,28)(15,19)(16,23)(20,32)(21,38)(22,34)(25,30)(26,35)(27,33)(29,36)(31,37)(39,40);(10,16)(11,20)(12,23)(13,14)(17,31)(18,35)(19,30)(21,37)(22,27)(24,28)(25,26)(29,32)(33,40)(34,38)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 42140);